Approximately Bisimilar Symbolic Models for Incrementally Stable Switched Systems

Abstract

International audienceSwitched systems constitute an important modeling paradigm faithfully describing many engineering systems in which software interacts with the physical world. Despite considerable progress on stability and stabilization of switched systems, the constant evolution of technology demands that we make similar progress with respect to different, and perhaps more complex, objectives. This paper describes one particular approach to address these different objectives based on the construction of approximately equivalent (bisimilar) symbolic models for switched systems. The main contribution of this paper consists in showing that under standard assumptions ensuring incremental stability of a switched system (i.e., existence of a common Lyapunov function, or multiple Lyapunov functions with dwell time), it is possible to construct a finite symbolic model that is approximately bisimilar to the original switched system with a precision that can be chosen a priori. To support the computational merits of the proposed approach, we use symbolic models to synthesize controllers for two examples of switched systems, including the boost dc-dc converter

Supervisory control of differentially flat systems based on abstraction

“…Unlike previous approaches [7], [3], [4], [5], [6], our technique provides a deterministic abstraction without requirements on the system's stability. This is a definite advantage when the complexity of the abstraction is a bottleneck.…”

“…Using Lemma 3.2 we conclude that, if φ [0,T (x(0),a)] (x(0), a) is δ-safe, then φ [0,T (χ(0),u)] (χ(0), u) is allowed by a supervisor σ C that ensures -safety with ≤ δ − τ (γ u − γ l )/2 − η. The quantity η = τ µ is defined in (5). From Lemma 3.6, we also know that a ≤ Cτ for some C > 0.…”

“…One of the greatest challenges in this approach resides in finding an efficient way to map the physical, continuous-time system onto a discrete event system, without losing too much structure along the way. Typical solutions involve defining a space and time discretization of the continuous-time model, restricting space and input sets so that the discretization is finite, and then proving an equivalence relation between the discretized system and a suitable discrete event system (as in [3], [4], [5], [6], [7]). The discrete event equivalent is then called a finite-state abstraction of the continuous-time system.…”

Finite controlled invariants for sampled switched systems

“…Sampled switched affine systems, as defined, e.g., in [10], is a subclass of affine hybrid systems [11]. A sampled switched affine system Σ is defined by a triple τ, U, F where: -τ ∈ R >0 is the time sampling parameter, -U = {1, .…”

“…Example 1. (Boost DC-DC converter) This example is taken from [4] (see also, e.g., [5,10,19]). This is a boost DC-DC converter with one switching cell (see Figure 1).…”